# Can someone give me some hints for this incline problem?

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1. Mar 18, 2016

### DavidFrank

• Member warned that HW questions must be posted in the HW sections
I am sorry, I generally would show some work but I have no clue on how to start off to find angle theta. The answer is given below and I have to get it. I don't know at all how to approach this problem and unfortunately I am not too strong on my trig identities either but I will try. May some please give me some hints?

2. Mar 18, 2016

### PeroK

Forget the complicated answer. Isn't moving horizontally a big clue?

3. Mar 18, 2016

### DavidFrank

Oh okay so Vfy = 0 and Vfx = Vfcos(0°) = Vf. Let me think about this some more, thank you.

4. Mar 18, 2016

### cnh1995

You need an equation relating θ and α. Since the ball is at maximum height, horizontal distance travelled is R/2. Do you know the relation between R, H and angle of projection(which is (θ+α) here)?

5. Mar 18, 2016

### cnh1995

That is not true. Vx is constant throughout and is equal to Vfcos(θ+α).
You need the relation between θ and α.

6. Mar 18, 2016

### haruspex

I believe David is using vf to mean the final velocity.

7. Mar 18, 2016

### haruspex

You are introducing new variables without defining them. I assume you are referring to usage in a range equation, in a form standard for you. David might not know this equation, or might not recognise those variable references.
I also guess PeroK is hinting at the same approach.
@DavidFrank , if cnh's post and PeroK's don't help, just apply the usual SUVAT equations for the horizontal and vertical motions. Do you know the SUVAT equations? (You should have listed them in the template.) can you figure out which ones to use?

8. Mar 18, 2016

### DavidFrank

That is so far all I have gotten. My attempt was to find "R" which is the incline distance of this projectile problem. I sort of ran into some issue when I tried finding Vi and substituting it into R = [2sin(θ)cos(θ+ α) * v.i^2] / [g * cos^2(α)] since the R cancels out.

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9. Mar 18, 2016

### DavidFrank

Thank you for your response haruspex, so far I have used the position formula and the Vf^2 = Vi^2 + 2a(Δx) formulas

10. Mar 18, 2016

### haruspex

I can confirm that the answer can be obtained from two of the equations you have developed.
Half way down, you have an equation in which the left hand side consists of sin(α)=. At the end, you have an equation where the left hand side consists of v02. Work with those.

11. Mar 19, 2016

### PeroK

I must admit I would never have thought of calculating R. The way I looked at it was that the point where it lands is $(x, y)$. I can calculate this from normal projectile motion (it's the top of the trajectory). And I know how this point relates to $\alpha$.

Perhaps this isn't the quickest way, but I ignored the answer and got:

$tan(\theta) = \frac{tan(\alpha)}{1+2tan^2(\alpha)}$

You could shoot for this if you like. Then, use a bit of trig manipulation to get the answer you need.

It looks better than to me than the answer given, not least because it is equivalent to:

$tan(\alpha + \theta) = 2tan(\alpha)$ (Which I find the most pleasing answer!)

12. Mar 19, 2016

### DavidFrank

I am sorry but I am stuck. Do you want to me to take a few steps back before I found "R" (the distance traveled on the incline) or should I keep on going by using Vfy2 = Viy2 + 2g(Yf - Yi)?

Because if I were to continue with the substitution of Vi^2 into "R" I'd get:

1 = [4 * sinΘ * sinα * cos(θ+α)] / [cos2α * sin2(θ+α)]

From here I feel willing to move all of the trig functions with (θ+α) to one side then separate them using the trig identity: cos (θ+α)=( cos θ)( cos α) – ( sin θ)( sinα) & sin (θ+α)=( sin θ)( cos α) + ( sinα)( cosθ) so that I can later group all the trig functions with θ and set it = to all the trig function with α, but it seems like a dead end. So can you please provide more hints on what you are saying before?

13. Mar 19, 2016

### cnh1995

Here's another way of doing it.
Horizontal displacement is x=R/2 and vertical displacement is H (R is for range and H is for maximum height).
If you take ratio of the expressions of R and H, you'll see
Rtan(Φ)=4H(a standard result!), where Φ is the angle of projection. In your problem, Φ=θ+α.
So,
Rtan(θ+α)=4H
Hence, xtan(θ+α)=2H where x=R/2.
tan(θ+α)=2H/x.
You can write the ratio H/x in terms of angle α(simple trigonometry) and do some trigonometric manipulations to get the final answer.

14. Mar 19, 2016

### DavidFrank

Looking at my teacher's notes, he did this:

cos(θ+α)*cosθ - sinθ*sin(θ+α) = 0
Thank you, I will give this a shot when I return home.

15. Mar 19, 2016

### haruspex

That equation is correct and does lead to the answer, though it is a struggle. I must have come a shorter way from the two equations I mentioned in post #10. (And as others have noted, there are easier ways to approach the whole question, but my preference is to help the OP finish their chosen path first.)
From the equation above, I got rid of all the cos and sin by writing t=tan theta, u= tan alpha. You can then get the whole equation in terms of t and u, using cos2=1/(1+tan2).
After some cancellation, you can get the whole thing into the form (....)2=0, affording an immediate simplification.
As a general formula? That would not be right. The left hand side equates to cos(2θ+α). Do you mean your teacher got the equations into this form?